Observations

The evidence from the Earth's bow shock and other solar system shocks is that collisionless shocks, at least for the observed parameter range, are not characterized by strong electron heating. Similarly, they are generally not responsible for strong electron acceleration, either in terms of fluxes or maximum energy. On the other hand, there is clear observational evidence for some shock acceleration of electrons to moderate energies at the bow shock and at interplanetary shocks. In both cases there is an electron foreshock populated by superthermal, energized electrons. The average thermal speed of the solar wind electron distribution is approximately 2000 km s–1 that the speed of electrons accelerated to only a few thermal energiesis much greater than the solar wind convection speed. Consequently the upstream edge of the electron foreshock can be taken as the tangent surface of magnetic field lines touching the shock surface. Accelerated ions, with their relatively slower speeds, are more affected by the solar wind convection and swept downstream. Thus, there is a region, downstream of the magnetic field tangent line surface and upstream of the ion foreshock, which can only be accessed by shock-accelerated electrons. This part of the electron foreshock, magnetically connected to the quasi-perpendicular shock, has the advantage, in terms of analysis, of not being disturbed by waves driven by accelerated ions. Direct particle observations at the electron foreshock at the Earth's bow shock gives evidence of shock acceleration from suprathermal energies to tens of keV. A review of electron foreshock observations and models is given by Fitzenreiter (1995).

Apart from direct measurements, the presence of energetic electrons can be inferred from the observations of Langmuir turbulence and other plasma waves driven via electron beams instabilities. At the Earth the electron foreshock is also a radio source at first and second multiples of the local plasma frequency, so-called fp and 2fp emission. Here the generation mechanism depends on electron beams which drive electrostatic Langmuir waves with frequency close to fpe via a beam instability. Some of the Langmuir wave energy is converted into fp and 2fp radiation by various linear or nonlinear wave–wave processes (Melrose, 1986). The fact that foreshock radio emission can be related to shock acceleration of electron beams gives the opportunity for remote sensing or characterizing shocks on larger scales than observable by a single spacecraft.

The study of shock waves in collisionless plasmas has a long history of over 50 years. That of shock waves in gas dynamics has roots which go back to the foundations of applied mathematics in the twentieth century, the nature of hyperbolic systems of equations and the physics of blast waves. Much of this early work was associated with military research, and some of the earliest work on shock waves in plasmas started from a similar background. However, in the early 1960s it was in space that truly collisionless shocks were first observed. With the advent of high-resolution space observations a fundamental challenge came into being: how can nonlinear collective processes replace the action of particle collisions and lead to thin shock waves in a collisionless plasma? In other words: how do collisionless shocks work?

With the growing exploration of space and better understanding of the plasma physics of the heliosphere, the importance of shocks has become evident. Shocks are formed around the planets in the supersonic flow of the solar wind; they are formed ahead of the impulsive flows of coronal mass ejections, and at the steady interaction regions between solar wind with different speeds; the entire region of the solar wind is defined by an outer boundary, the solar wind termination shock, where the flow transitions to subsonic as it comes into balance with the interstellar medium. In parallel to our increasing understanding of solar system shocks, it has become obvious that shock waves will arise in many other astrophysical systems, and that often the physics will be dominated by collisionless processes. A widely cited example is the shock wave driven by a supernova remnant; such shocks are understood to be vital for explaining the majority of cosmic ray acceleration.

In writing this book we had three aims in sight. We felt there was a need for a graduate level textbook that brought together the physics of collisionless shocks as found in the heliosphere, with an emphasis on the theoretical underlying physics of shock waves in plasmas.

Observations of diffuse ions at heliospheric shocks

Virtually all shocks observed either in situ or indirectly in the solar system are accompanied by energetic particles, i.e., protons, heavy ions and electrons, with energies up to 1000MeV and higher. The most intense solar energetic particle (SEP) events are produced by acceleration at interplanetary shock waves driven by coronal mass ejections (CMEs). The intensity–time profiles of electrons and protons in CME as sociated particle events have usually a fast rise and a decay phase extending over several days. The shock passage at 1 au occurs early in the decay phase and is often accompanied by a peak in the lower-energy ions (˜1MeV). Since these events are long-lasting (several days) they have been termed gradual events, as opposed to impulsive events of a duration of approximately a few hours. The latter are associated with impulsive X-ray flares and type III radio bursts. X-ray emission from flares indicates plasma heating to temperatures of order ten million kelvin, and solar type III radio bursts imply the impulsive production of electrons travelling at 0.1–0.25 times the speed of light. Acceleration at the CME driven shocks populate magnetic field lines over a broad range of longitudes, while the impulsive event sare generally detected when the observer is magnetically connected to the flare site. Figure 6.1a shows the intensity–time profiles of particle fluxes (one electron and three proton energy channels) in a ‘pure’ gradual event accompanied by a CME and Fig. 6.1b shows an impulsive event due to a series of flares (Reames, 1999).

There is clear evidence that energetic ions are accelerated at interplanetary shocks out of the solar wind thermal population. Gosling et al. (1981) have transformed the ion distribution function measured over a wide energy range (10 eV to 1.6MeV) behind an interplanetary shock into the solar wind frame. Figure 6.2 shows a cut of the distribution function along the Sun–Earth line which demonstrates that the suprathermal distribution emerges smoothly out of the solar wind thermal distribution, which can be taken as evidence that the solar wind ions are accelerated to high energies.

Introduction

For most of this book we have treated collisionless shocks as planar infinitely extended discontinuities with upstream plasma parameters characteristic for the solar wind, possibly corrugated by instabilities. However, all shocks in the solar system are curved and consequently the angles θBn, between the magnetic field and shock normal, and θVn, between the upstream plasma flow and shock normal, change along the shock surface. Ions backstreaming from a certain part of a shock can excite upstream waves which can be convected into other parts of the shock, where they may influence the shock structure. This is of particular importance for planetary bow shocks, where the region upstream of the quasi-perpendicular part of the shock can interact further downstream with the quasi-parallel part of the shock. In addition, the decrease in θVn, toward the flanks results in considerable variation of the shock strength along the bow shock. In Section 9.2 we will discuss results on global bow shock simulations and, in Section 9.3, the influence of bow shock curvature on upstream escaping electrons.

In Chapter 6 it was described how particles can be accelerated to very high energies by the converging flows in shocks. The maximum energy is ultimately only limited by the finite spatial extent of the shock or by the maximum time available for the diffusive shock acceleration mechanism to work. The pressure in the accelerated particles in the upstream region can become very high and can even exceed the thermal particle pressure. This will lead to considerable modification of the shock structure. Although probably not important for shocks in the solar system, we will briefly outline in Section 9.4 the methods which have been used in order to treat shocks mediated by energetic particles which have, in turn, been produced by these shocks. As stated before, we are not so much concerned with the energetic particles (cosmic rays), but with the influence these may have on the shock structure.

Before reaching the heliopause where the solar wind pressure is balanced by the pressure of the local interstellar medium, the solar wind is slowed from a super-fast mode speed flow to a sub-fast mode speed flow at the heliospheric termination shock. The structure of the heliospheric termination shock might not be expected to be so different from other shocks in the solar system, if the upstream medium were much like the solar wind in the inner heliosphere. But the solar wind in the region of the termination shock carries with it a high percentage of pickup protons.

Introduction

The accumulation of observations of the quasi-perpendicular, high Mach number terrestrial bow shock has led to a consensus view that the main features of shock heating can be explained by a time-stationary macroscopic model for the shock fields: specular reflection of some fraction of the ions produces the bulk of the heating required by the shock jump conditions, and the macroscopic profile of electric and magnetic field adjusts to provide the required reflected fraction; the electron heating is mediated by the cross-shock electric potential in the de Hoffmann–Teller frame. However, as discussed in Chapter 3, although the emphasis is on the macroscopic fields, small-scale fluctuations are vital for a convincing and self-consistent shock model, even on a conceptual basis. Wave–particle coupling in turbulent fluctuations is required to explain ion isotropization, the infilling of the electron distribution function, and, perhaps most importantly, time irreversibility and entropy increase at the shock, all of which a static macrostructure is incapable of providing.

In this chapter we discuss the various sources of microstructure at the quasi-perpendicular shock, concentrating on the high Mach number regime. Of course, there is a certain degree of arbitrariness in the distinction between macro-and microstructure. At one extreme one could assign the heating and dissipation at the shock entirely to the effects of inhomogeneous fluctuations, and then the macro-scopic structure would take whatever form was dictated by a particular choice of spatial and temporal averages. However, the usual approach is to assume that there exists a well-defined underlying macrostructure, which can be related to a multi-fluid model of reflected and transmitted ions, producing the canonical foot–ramp–overshoot structure in the magnetic field. Once the macroscopic structure has been removed, the ensemble of plasma and field fluctuations which remains can give rise to anomalous dissipation and particle heating, acceleration of particles, shock ramp broadening and temporal/spatial changes of the shock structure.

Introduction

Surveying the data from an Earth-orbiting spacecraft the differences between crossings of the quasi-parallel and quasi-perpendicular bow shock are usually easy to distinguish by their characteristic appearance in the magnetic field. Compared with the extended, turbulent transition of the quasi-parallel shock, the abrupt transition of the quasi-perpendicular shock resembles the idealized MHD shock discontinuity. The apparent simplicity of the quasi-perpendicular shock has produced a large body of observational work, which has focused on the average time-steady structure of the shock transition – the macrostructure of the shock. For most solar wind conditions the Mach number at the Earth's bow shock (and similarly for other solar system shocks) is strong enough to be greater than the critical Mach number found in MHD when resistivity is the dominant dissipation mechanism (see Section 2.4). At such supercritical shocks it is found that ion reflection is a characteristic feature, even though it is a consequence of the kinetics of particles in the shock fields. In this chapter we describe the macrostructure of supercritical quasi-perpendicular shocks, as found at the Earth's bow shock. This is by no means the complete picture, since, as well as the macrostructure, fluctuations are also present across the entire range of plasma time scales, from below the ion cyclotron frequency to above the electron plasma frequency. In Chapter 4 we discuss the microstructure of the quasi-perpendicular shock and its various causes.

The bow shock is observed when it passes over a spacecraft, since the spacecraft motion is generally much slower than that of the bow shock, in the Earth frame. The bow shock position varies in response to changes in the solar wind, and moves with a velocity up to several tens of kilometres per second, both outwards from the Earth and inwards towards the Earth. In a time series of data the bow shock is seen either as a transition from solar wind to magnetosheath (‘inward’ crossing) or from magnetosheath to solar wind (‘outward’ crossing). The quasi-perpendicular shock is characterized as ‘abrupt’, but what is actually observed in the time series will depend on the physical size of the transition, the sample cadence or averaging (i.e, the time resolution) of the data, and the bow shock speed relative to the spacecraft.

Introduction

A shock is an abrupt transition between supersonic and subsonic flows. The best-known example is that formed by an obstacle, such as an aircraft, travelling through air faster than the speed of sound. For the aircraft to move forward the air ahead of it must be diverted around it, and there has to be a layer of subsonic flow adjacent to the obstacle. If this were not the case the influence of the pressure force, which propagates away from the obstacle at the sound speed, would be swept downstream and could not affect the flow ahead of the obstacle. Relative to the obstacle the distant flow is supersonic, so there has to be a transition to the subsonic flow close to the obstacle.

Considering such transitions within the framework of gas dynamics leads to the study of discontinuous, or near-discontinuous, solutions which must satisfy governing equations such as conservation of mass, momentum and energy. These solutions represent abrupt, well-defined changes in flow state, with so called ‘jumpconditions’ which describe the supersonic to subsonic transition. Two defining qualities can be extracted from this framework. Firstly, at a shock the flow speed changes, but also the temperature increases due to dissipation, so that the shock mediates a transfer of bulk kinetic energy in the upstream flow into thermal energy downstream. The presence of dissipation means that the change of state at a shock corresponds to an entropy increase, and it is irreversible. Secondly, by their nature,shock solutions are fundamentally nonlinear, since not only do they accomplish achange of state, but they can also be thought of as a ‘wave’ whose propagation speed is determined by the supersonic flow speed, i.e., faster than any small-amplitude linear wave. It is nonlinearity which leads to wave steepening at the shock, and the importance of discontinuous solutions.

Shocks have been explored in laboratory plasmas since the 1950s, but the discovery of shocks in interplanetary space, and the confirmation that they are relatively stable structures, has been one of the major advances of space physics. Neverthe-less, the study of shocks in plasmas which are essentially collisionless poses some fundamental problems.